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Pisot sequences E(6,10), P(6,10).
1

%I #27 Mar 10 2020 10:27:40

%S 6,10,17,29,49,83,141,240,409,697,1188,2025,3452,5885,10033,17105,

%T 29162,49718,84764,144514,246382,420057,716156,1220976,2081645,

%U 3549002,6050703,10315860,17587538,29985042,51121581,87157325,148594765,253339627,431919433

%N Pisot sequences E(6,10), P(6,10).

%H Colin Barker, <a href="/A020718/b020718.txt">Table of n, a(n) for n = 0..1000</a>

%H Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, <a href="https://arxiv.org/abs/1609.05570">Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences</a>, arXiv:1609.05570 [math.NT], 2016.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,0,1,-1).

%F G.f.: (-4*x^5+x^4+x^3-3*x^2-2*x+6)/((1-x)*(1-x-x^2-x^5)) (conjectured). - _Ralf Stephan_, May 12 2004

%F a(n) = 2*a(n-1)-a(n-3)+a(n-5)-a(n-6) for n>5 (conjectured). - _Colin Barker_, Jun 05 2016

%F Theorem: E(6,10) satisfies a(n) = 2 a(n - 1) - a(n - 3) + a(n - 5) - a(n - 6) for n >= 6. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger. This shows that the above conjectures are correct. - _N. J. A. Sloane_, Sep 10 2016

%t LinearRecurrence[{2, 0, -1, 0, 1, -1}, {6, 10, 17, 29, 49, 83}, 30] (* _Jinyuan Wang_, Mar 10 2020 *)

%Y See A008776 for definitions of Pisot sequences.

%K nonn

%O 0,1

%A _David W. Wilson_